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Full text of "Treatise On Analysis Vol-Ii"

21    PRODUCT OF MEASURES        241

Let j?, q be such that \<>p< q< +00. Show that, if U is of weak type (p}p) and also
of weak type (q, q\ then U is of type (r, r) (Section 13.17, Problem 7) for all r such
that p<r<q (Marcinkiewicz's interpolation theorem). (For each /e -^(X, ju,) and
a > 0, let/ be the function which is equal to f(x) if \f(x)\ > a, and zero otherwise,
and put/ = /-/. Show that, for each / > 0,

Integrate from 0 to oo and use Problem 1 .)
R*(/) =   sup |Rk(/)|,